Learning Proximal Operators to Discover Multiple Optima
This addresses a ubiquitous problem in optimization for applications such as object detection, offering a novel end-to-end method that improves over ad hoc heuristics.
The paper tackles the challenge of finding multiple solutions in non-convex optimization by learning a proximal operator from training problems, enabling quick recovery of multiple local minima from initial guesses and generalization to unseen problems like object detection, with effectiveness demonstrated through an exhaustive benchmark.
Finding multiple solutions of non-convex optimization problems is a ubiquitous yet challenging task. Most past algorithms either apply single-solution optimization methods from multiple random initial guesses or search in the vicinity of found solutions using ad hoc heuristics. We present an end-to-end method to learn the proximal operator of a family of training problems so that multiple local minima can be quickly obtained from initial guesses by iterating the learned operator, emulating the proximal-point algorithm that has fast convergence. The learned proximal operator can be further generalized to recover multiple optima for unseen problems at test time, enabling applications such as object detection. The key ingredient in our formulation is a proximal regularization term, which elevates the convexity of our training loss: by applying recent theoretical results, we show that for weakly-convex objectives with Lipschitz gradients, training of the proximal operator converges globally with a practical degree of over-parameterization. We further present an exhaustive benchmark for multi-solution optimization to demonstrate the effectiveness of our method.